Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Defining Reasoning and Proof
|Introduction | Deductive Reasoning | Proof by Contradiction | Inductive Reasoning | Proof by Mathematical Induction | The Teachers' Role | Your Journal|
On the topic of induction and deduction, there is one other term to introduce and define, proof by "mathematical induction" which constitutes a special case. This is, in fact, a deductive method in which a proposition to be proved, generally about the natural numbers, is shown to be true for one case and a method is established by which the same reasoning about this case is applicable to the next (n + 1) case. So therefore the proposition holds. Frequently a row of dominoes or a ladder is invoked as a handy metaphor for mathematical induction. If you can get on the ladder, and then move from one rung to the next, you have a method to climb the whole ladder. Similarly, if you know that when you knock down one domino standing in row that it will knock the next one and so forth, then all you have to do is knock the first domino and you have floored the whole row.
This method could be used to prove the proposition that was one element of the Explore activity. For any positive integer n, show that 1 + 2 + 3 + . . . + n = n (n + 1) / 2.
An inductive proof might go as follows:
Let our conjecture, P (n) be that 1 + 2 + 3 + . . . + n = n (n + 1) / 2.
For the case n = 1, P (1) = 1, we have a valid instance.
Now let us look at another case, P (k), and show that if our rule holds for P (k), then this implies it is true for P (k + 1).
P (k) = 1 + 2 + 3 + . . . + k (1)
P (k + 1) = 1 + 2 + 3 + . . . + k + (k + 1) (2)
Substituting (2) in our formula, we get P (k + 1) = 1 + 2 + 3 + . . . + k + (k + 1) = [(k + 1) (k + 2)] / 2 and this is what we must prove.
We have established that the first k terms can be represented as k (k +1) / 2. So substituting this expression, gives us P (k + 1) = k (k + 1) / 2 + (k + 1) and this can be simplified algebraically to (k + 1) (k + 2) / 2, the conjecture we were trying to prove.
Mathematical induction is a technique that students should be familiar with by the end of high school student. They should also be able to distinguish this specialized usage of the term induction from the more general sense of inductive reasoning discussed above.
|Teaching Math Home | Grades 9-12 | Reasoning and Proof | Site Map | © ||